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Solenoid
created by a seven-loop solenoid (cross-sectional view) described using s.}} A solenoid is a coil wound into a tightly packed . The term was invented in 1823 by to designate a helical coil. As a technical term in the study of , a solenoid is a coil that is "pipe-like" in the sense that its length is substantially greater than its diameter. In practice, the coil is often wrapped around a lic core, which produces a uniform in a volume of space (where some experiment might be carried out) when an is passed through it. A solenoid is a type of the purpose of which is to generate a controlled magnetic field. If the purpose of the solenoid is instead to impede changes in the electric current, a solenoid can be more specifically classified as an rather than an electromagnet. The solenoid is not necessarily straight, for example, 's electromagnet of 1824 consisted of a solenoid bent into a horseshoe shape. In , the term may also refer to a variety of devices that convert into linear motion. The term is also often used to refer to a , which is an integrated device containing an electromechanical solenoid which actuates either a or valve, or a solenoid switch, which is a specific type of that internally uses an electromechanical solenoid to operate an electrical switch; for example, an , or a linear solenoid, which is an electromechanical solenoid. s, a type of electronic-mechanical locking mechanism, also exist. Infinite continuous solenoid An infinite solenoid is a solenoid with infinite length but finite diameter. Continuous means that the solenoid is not formed by discrete finite-width coils but by infinitely many infinitely-thin coils with no space between them; in this abstraction, the solenoid is often viewed as a cylindrical sheet of conductive material. Inside labeled a'', ''b and c''. Integrating over path ''c demonstrates that the magnetic field inside the solenoid must be radially uniform.}} The inside an infinitely long solenoid is homogeneous and its strength neither depends on the distance from the axis, nor on the solenoid's cross-sectional area. This is a derivation of the around a solenoid that is long enough so that fringe effects can be ignored. In Figure 1, we immediately know that the flux density vector points in the positive z'' direction inside the solenoid, and in the negative ''z direction outside the solenoid. We confirm this by applying the for the field around a wire. If we wrap our right hand around a wire with the thumb pointing in the direction of the current, the curl of the fingers shows how the field behaves. Since we are dealing with a long solenoid, all of the components of the magnetic field not pointing upwards cancel out by symmetry. Outside, a similar cancellation occurs, and the field is only pointing downwards. Now consider the imaginary loop c'' that is located inside the solenoid. By , we know that the of '''B' (the magnetic flux density vector) around this loop is zero, since it encloses no electrical currents (it can be also assumed that the circuital passing through the loop is constant under such conditions: a constant or constantly changing current through the solenoid). We have shown above that the field is pointing upwards inside the solenoid, so the horizontal portions of loop c'' do not contribute anything to the integral. Thus the integral of the up side 1 is equal to the integral of the down side 2. Since we can arbitrarily change the dimensions of the loop and get the same result, the only physical explanation is that the integrands are actually equal, that is, the magnetic field inside the solenoid is uniform. Outside A similar argument can be applied to the loop ''a to conclude that the field outside the solenoid is constant. An intuitive argument can also be used to show that the flux density outside the solenoid is actually zero. Magnetic field lines only exist as loops, they cannot diverge from or converge to a point like electric field lines can (see ). The magnetic field lines follow the longitudinal path of the solenoid inside, so they must go in the opposite direction outside of the solenoid so that the lines can form a loop. However, the volume outside the solenoid is much greater than the volume inside, so the density of magnetic field lines outside is greatly reduced. Now recall that the field outside is constant. In order for the total number of field lines to be conserved, the field outside must go to zero as the solenoid gets longer. Quantitative description Applying to the solenoid (see the right figure) gives us : B l= \mu_0 N I, where B is the , l is the length of the solenoid, \mu_0 is the , N the number of turns, and I the current. From this we get : B = \mu_0 \frac{N I}{l}. This equation is valid for a solenoid in free space, which means the permeability of the magnetic path is the same as permeability of free space, μ0. If the solenoid is immersed in a material with relative permeability μr, then the field is increased by that amount: : B = \mu_0 \mu_{\mathrm{r}} \frac{N I}{l}. In most solenoids, the solenoid is not immersed in a higher permeability material, but rather some portion of the space around the solenoid has the higher permeability material and some is just air (which behaves much like free space). In that scenario, the full effect of the high permeability material is not seen, but there will be an effective (or apparent) permeability μ''eff such that 1 = ''μ''eff = ''μ''r. The inclusion of a core, such as , increases the magnitude of the magnetic flux density in the solenoid and raises the effective permeability of the magnetic path. This is expressed by the formula : B = \mu_0 \mu_{\mathrm{eff}} \frac{N I}{l} = \mu \frac{N I}{l}, where ''µ''eff is the effective or apparent permeability of the core. The effective permeability is a function of the geometric properties of the core and its relative permeability. The terms relative permeability (a property of just the material) and effective permeability (a property of the whole structure) are often confused; they can differ by many orders of magnitude. For an open magnetic structure, the relationship between the effective permeability and relative permeability is given as follows: : \mu_\mathrm{eff} = \frac{\mu_r}{1+k(\mu_r -1)}, where ''k is the demagnetization factor of the core. Inductance As shown above, the magnetic flux density B within the coil is practically constant and given by : B = \mu_0 \frac{NI}{l}, where µ''0 is the , N the number of turns, I the current and l the length of the coil. Ignoring end effects, the total through the coil is obtained by multiplying the flux density B by the cross-section area A : : \Phi = \mu_0 \frac{NIA}{l}. Combining this with the definition of : L = \frac{N \Phi}{I}, the inductance of a solenoid follows as : L = \mu_0 \frac{N^2A}{l}. A table of inductance for short solenoids of various diameter to length ratios has been calculated by Dellinger, Whittmore, and Ould. This, and the inductance of more complicated shapes, can be derived from . For rigid air-core coils, inductance is a function of coil geometry and number of turns, and is independent of current. Similar analysis applies to a solenoid with a magnetic core, but only if the length of the coil is much greater than the product of the relative of the magnetic core and the diameter. That limits the simple analysis to low-permeability cores, or extremely long thin solenoids. The presence of a core can be taken into account in the above equations by replacing the magnetic constant ''µ0 with µ'' or ''µ0µr, where µ'' represents permeability and ''µr . Note that since the permeability of materials changes with applied magnetic flux, the inductance of a coil with a ferromagnetic core will generally vary with current. References Category:Electricity